Accurate Numerical Integration of Perturbed Oscillatory Systems in Two Frequencies

ABSTRACT

Highly accurate long-term numerical integration of nearly oscillatory systems of ordinary differential equations (ODEs) is a common problem in astrodynamics. Scheifele’s algorithm is one of the excellent integrators developed in the past years to take advantage of special transformations of variables such as the K-S set. It is based on using expansions in series of the so-called G-functions, and generalizes the Taylor series integrators but with the remarkable property of integrating without truncation error oscillations in one basic known frequency. A generalization of Scheifele’s method capable of integrating exactly harmonic oscillations in two known frequencies is developed here, after introducing a two parametric family of analytical ϕ-functions. Moreover, the local error contains the perturbation parameter as a factor when the algorithm is applied to perturbed problems. The good behavior and the long-term accuracy of the new method are shown through several examples, including systems with low- and high-frequency constituents and a perturbed satellite orbit. The new methods provide significantly higher accuracy and efficiency than a selection of well-reputed general-purpose integrators and even recent symplectic or symmetric integrators, whose good behavior in the long-term integration of the Kepler problem and the other oscillatory systems is well stated in recent literature.

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INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
      G.1.7 Ordinary Differential Equations

General Terms:
Algorithms

Keywords:
Numerical solutions of ODEs, highly oscillatory solutions, long-term integration, perturbed oscillators